The depth of a permutation

Details The depth of a permutation The depth of a permutation

2012-02-21

arxiv.org/abs/1202.4765v1

Mathematics

Combinatorics

23 pages

Scientific paper

For the elements of a Coxeter group, we present a statistic called depth, defined in terms of factorizations of the elements into products of reflections. Depth is bounded above by length and below by reflection length. In this article, we focus on the case of the symmetric group, where we show that depth is equal to sum_i max{w(i)-i, 0}. We characterize those permutations for which depth equals length: these are the 321-avoiding permutations (and hence are enumerated by the Catalan numbers). We also characterize those permutations for which depth equals reflection length: these are permutations avoiding both 321 and 3412 (also known as boolean permutations, which we can hence also enumerate). In this case, it also happens that length equals reflection length, leading to a new perspective on a result of Edelman.

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